Tri-axial accelerometers and magnetometers are widely used in oil and gas well characterization to measure the components of gravitational g and the earth magnetic field intensity B in a tool coordinate system. By convention, the tool z-axis is the long axis of the tool, which corresponds to the borehole axis. The magnetometers and accelerometers are calibrated in a laboratory for scale factor, bias, and misalignment. These sensors are calibrated for temperatures up to their specifications (e.g., up to 150° C., 175° C., 200° C., etc). In addition, the validity of calibration is also periodically checked at room temperature.
While such laboratory calibration methods of sensors' scale factor, bias, misalignment, temperature work reasonably well downhole, these sensors still suffer from systematic errors due to the aging effect of sensors (e.g., sensor characteristic change over months from high-temperature and high-shock exposures), internal physical component change/damage, etc. Generally, such systematic errors could be observed in the field in-between calibration periods (every 2-3 months to 6 months). For example, bias errors greatly affect the results of accelerometers and magnetometers at near-vertical inclination and/or at directions near the magnetic dip axis. Axial sensor misalignment affects the measurement consistency and precision (resulting in misalignment errors) at near-horizontal inclination and/or at directions close to magnetic east/west, where axial accelerometer and/or magnetometer reading are very small. The aforementioned systemic errors, bias errors, and misalignment errors are referred to as calibration error throughout this disclosure.